Short-range repulsive force

For interactions between neutrals, both long- and short-range interactions have to be accounted for. At long range van der Waals dispersion forces or incipient chemical bonding are operative, and at short range the Pauli exclusion principle ensures a strong repulsive force. Empirical potentials have been extensively used to describe binary 

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The forces which bring about adsorption always include dispersion forces, which are attractive, together with short-range repulsive forces. In addition, there will be electrostatic (coulombic) forces if either the solid or the gas is polar in nature. Dispersion forces derive their name from the close connection between their origin and the cause of optical dispersion. First... 

An expression for the short-range repulsive force (which arises from the interpenetration of the electron clouds of the two atoms) can also be derived from quantum-mechanical considerations" as... 

FIG. 16-4 Depictio ns of surface excess F- Top The force field of the sohd concentrates component near the surface the concentration C is low at the surface because of short-range repulsive forces between adsorbate and surface. Bottom Surface excess for an imagined homogeneous surface layer of thickness Axf... 

Nevertheless, previous developments and some of our results prove that the structural properties of several systems with short-range repulsive forces are straightforwardly and sufficiently accurately given by ROZ integral equations. Thermodynamic properties are much more difficult to describe. Reliable tools exist to obtain thermodynamics at high temperatures or for states far from phase transitions. Of particular importance, and far from being solved, are the issues related to phase transitions in partly quenched systems, even for simple models with attractive interactions. It seems that the results obtained by Kierlik et al. [27], may serve as a helpful reference in this direction. 

Physically the independence reflects the fact that dephasing is performed by weak long-range interactions, and rotational relaxation results mainly from short-range, repulsive forces. In other words the rotational state is changed solely when the distance between molecules becomes rather short, while the phase is frustrated in all cases and the contribution of frontal collisions is not so significant. 

While Debye and HUckel recognized the short-range repulsive forces between ions by assuming a hard-core model, the statistical mechanical methods then available did not allow a full treatment of the effects of this hard core. Only the effect on the electrostatic energy was included—not the direct effect of the hard core on thermodynamic properties. 

Although not normally adopted in geochemistry, this form best takes into account the functional form of short-range repulsive forces (cf eq. 1.54). 

Short-range repulsive forces are a direct result of the Pauli exclusion principle and are thus quantum mechanical in nature. Kitaigorodskii (1961) has emphasized that such short-range repulsive forces play a major role in determining the packing in molecular crystals. The size and shape of molecules is determined by the repulsive forces, and the molecules pack as closely as is permitted by these forces. 

At extremely short distances, for example, zo < 3 A, the repulsive force becomes dominant. It has a very steep distance dependence. The tip-sample distance is virtually determined by the short-ranged repulsive force. By pushing the tip farther toward the sample surface, the tip and sample deform accordingly. 

In actuality, molecules in a gas interact via long-ranged attractions and short-range repulsive forces. An interaction potential energy function is used to describe these forces as a function of intermolecular distance and orientation. This section introduces two commonly used interaction potential energy functions. 

Evaw = s[% 12 - 2( )6] where rm n is the distance between two atoms i and j when the energy is at a minimum e and r is the actual distance between the atoms. This equation is known as the Lennard-Jones 6-12 potential. The ( )6 term in this equation represents attractive forces, whilst the ( )12 term represents short range repulsive forces between the atoms. 

In pure liquids, short-range repulsive forces are responsible for most of the dephasing. The viscoelastic theory describes the interaction of these forces with the diffusive dynamics of the liquid (Section IV.D). The resulting frequency modulation is in the fast limit in low-viscosity liquids but can reach the slow-modulation limit at higher viscosities. This type of dephasing was seen in supercooled toluene (Section IV.C). 

Arrays of ions tend to maximize the net electrostatic attraction between ions, while minimizing the repulsive interactions. The former ensures that cations are surrounded by anions, and anions by cations, with the highest possible coordination numbers. In order to reduce repulsive forces, ionic solids maximize the distance between like charges. At the same time, unlike charges cannot be allowed to get too close, or short-range repulsive forces will destabUize the stmcmre. The balance between these competing requirements means that ionic sohds are highly symmetric stmctures with maximized coordination numbers and volumes. 

Interpreting bulk properties qualitatively on the basis of microscopic properties requires only consideration of the long-range attractive forces and short-range repulsive forces between molecules it is not necessary to take into account the details of molecular shapes. We have already shown one kind of potential that describes these intermolecular forces, the Lennard-Jones 6-12 potential used in Section 9.7 to obtain corrections to the ideal gas law. In Section 10.2, we discuss a variety of intermolecular forces, most of which are derived from electrostatic (Coulomb) interactions, but which are expressed as a hierarchy of approximations to exact electrostatic calculations for these complex systems. 

The solution depends on the interaction forces, including long range attractive interactions, contact and short range repulsion forces, JKR forces, etc. The relevant parameters include the (driving) frequency, the phase, and the amplitude of the oscillation (Fig. 1.12). 

As a result of assumption (ii) above, it will be shown that the value of can be expressed as the sum over the contributions of all the atoms present in the molecule, with these contributions being induced by the potential field of the surface. It should be noted here that although the interactions can be divided into two major classes, via pair or atom-atom interactions, such as dispersion and short-range repulsion forces, and an essentially non-pair interaction, that due to polarization, they can both be treated in a similar fashion within the framework of the proposed method. 

Case II refers to situations where the particle-wall interactions are purely repulsive. The particles are separated from the wall by a thin layer of solvent, even in the absence of any motion. Slip is thus possible for very slow flows, indicating that the sticking yield stress is vanishingly small. The residual film thickness for weak flows corresponds to a balance between the osmotic forces and the short-range repulsive forces, independently of any elastohydrodynamic contribution. This is clearly reflected in Fig. 16c, d, where we observe that the particle facet is nearly flat and symmetric. Since tire pressure in the leading and rear regions of the facet are equal and opposite, the lift force is very small. The film thickness, which is set by the balance of the short-range forces, is constant so that the stress/velocity relationship is linear. 

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